I’ve recently made determinations of the dead time constant via the ratio method using the Mo Lα and Mo Lβ

_{3} peaks on uncoated, recently cleaned Mo metal in the same manner as in previous runs, though I increased the number of calculated ratios within a given set (60 in the first and 70 in the second). I chose the Lβ

_{3} peak instead of Lβ

_{1} because I wanted to obtain a ratio of peak count rates between about 15 and 20 on a given spectrometer such that the source of noticeably non-linear correction within each calculated ratio would be accounted for essentially solely by the Mo Lα measurement. While Si Kα/Kβ might have worked for this purpose just as well, the ratio of peak heights is even greater (so that it would have been unreasonably time-consuming to obtain an adequate number of Si Kβ counts to keep counting error low). Unfortunately, Ti Kβ cannot be accessed on PETH, which I’d forgotten about. For the first measurement set on Mo, I collected data at PCD currents ranging from 5 to 600 nA, and, for the second set, from 5 to 700 nA. The maximum observed count rate (for Mo Lα on PETH at

*I*_{PCD} = 700 nA) was roughly 240 kcps. On channel 2/PETL at the same current, observed Mo Lβ

_{3} count rate was about 13 kcps.

Notation for the two measurement sets is as follows:

*N’*_{11} = measured Mo Lα count rate on channel 2/PETL

*N’*_{21} = measured Mo Lβ

_{3} count rate on channel 3/PETJ

*N’*_{31} = measured Mo Lβ

_{3} count rate on channel 5/PETH

*N’*_{12} = measured Mo Lβ

_{3} count rate on channel 2/PETL

*N’*_{22} = measured Mo Lα count rate on channel 3/PETJ

*N’*_{32} = measured Mo Lα count rate on channel 5/PETH

I was hoping that my calculated dead times would be virtually identical to those I’ve previously determined, as I no longer believe that X-ray counter dead time varies systematically with X-ray energy. (But maybe I'm wrong considering the pattern possibly emerging in the calculated dead times.) Perhaps my PHA settings weren’t quite right? If so, considering that PHA shifts should be smaller for lower X-ray energies, my most recent determinations on Ti and Mo should be the most accurate. The age of the counters could be a factor as well, as anomalies certainly are present in the pulse amplitude distributions, at least under certain conditions. I examined these distributions carefully, though, across a wide range of count rates, and I don’t think I made any serious errors. Maybe I’ll run through the whole process again on Cu or Fe and see if I get the same results as before. Currently, I have dead time constants for my channels 2, 3, and 5 set at 1.45, 1.40, and 1.40 μs, respectively, but I might eventually lower the values for channels 2 and 3 a little – channel 5 has been more consistent. At any rate, the values for Mo aren’t drastically lower and are most similar to those obtained from Ti. Continuing the same pattern as before, channel 2 gives the largest dead time constant using the ratio method:

Mo Lα/Mo Lβ_{3} ratio method [μs]:

channel 2/PETL: 1.38, 1.43 (Ti: 1.43, 1.46; Fe: 1.44, 1.48; Cu: 1.50, 1.46)

channel 3/PETJ: 1.33 (Ti: 1.37; Fe: 1.41; Cu: 1.45)

channel 5/PETH: 1.37 (Ti: 1.42; Fe: 1.41; Cu: 1.42)

Current-based method, Mo Lα [μs]:

channel 2/ PETL: 1.27

channel 3/ PETJ: 1.17

channel 5/ PETH: 1.27

Current-based method, Mo Lβ_{3} [μs]:

channel 2/ PETL: 0.18

channel 3/ PETJ: -1.15

channel 5/ PETH: 0.20

The apparent picoammeter anomaly was somewhat less pronounced than it was during my measurements on Ti. Its somewhat lower magnitude accounts for the slightly larger Mo Lα dead time values (compared to those for Ti Kα) calculated using the current-based method below 85 kcps (measured):

Channel 3 shows the ugliness a little better due to its low count rate at high current:

As an aside, I’d like to emphasize that operating sealed Xe counters routinely at count rates greater than tens kcps constitutes abuse of them and shortens their useful lifespans, even if anode bias is kept relatively low (~1600-1650 V).

Below I’ve fit both a straight line and a parabola (dashed curve) to the uncorrected data for

*N’*_{12}/

*N’*_{32} (where

*N’*_{12} corresponds to Mo Lβ

_{3} on channel 2/PETL and

*N’*_{32} to Mo Lα on channel 5/PETH). Although the parabola fits the data very nicely (R

^{2} = 0.9997), easy calculation of

*τ*_{1} and

*τ*_{2} relies on use of a correction function linear in

*τ*, such as

*N’*/

*N* = 1 –

*τN’*. Although the two dead time constants cannot be extracted readily from the second degree equation, since Mo Lβ

_{3} count rate on channel 2/PETL does not exceed 13 kcps (at

*I*_{PCD} = 700 nA), essentially all non-linear behavior is contributed to the ratio by Mo Lα on channel 5/PETH, and so it alone accounts for the quadratic term. The regression line (R

^{2} = 0.991) that provides the linear approximation at measured count rates below ~85 kcps is roughly tangent to the parabola at

*N’*_{12} = N

_{12} = 0, and so both intersect the vertical axis at or close to the true ratio,

*N*_{32}/

*N*_{12} (subject to counting error, of course). The size of the parabola is such that, below ~85 kcps, it is approximated very well by a straight line (considering counting error). Further, in truth, the dead time constant determined in the effectively linear correction region should work just as well in equations of higher degree or in any equation in which the true count rate,

*N*, is calculated (realistically) as a function of the measured count rate,

*N’*. If the dead time constant can be calculated simply and accurately in the linear correction region on a single material (and only at the peak), then there is no need to make the process more complicated if only to avoid additional propagated counting error and potential systematic errors.

And here I’ve magnified the above plot to show the region of relatively low count rates: